Download Measuring Size in Astronomy

PARALLAX
“the apparent shift or
movement of a nearby object against a
distant background, when viewed from
two different positions”
 Definition:
Examples of parallax

Thumb in front of the eyes
 Speedometer (not digital) viewed by the
driver vs. the passenger
 An object in the room when viewed from
the RHS of the room vs. the LHS
 A flagpole when viewed from opposite ends
of a sidewalk
Vocabulary of parallax

Baseline
 Parallax shift or parallax angle
 Distance
 Small angle formula
 Arcseconds
Geometry of parallax
1
C
d1
B
E
Distant
Background
< p2
< p1
Baseline
F
A
d2
D
2
Conclusions

Nearby objects have a greater parallax shift
and a larger parallax angle than more distant
objects.
 Conversely: If an object A has a larger
parallax shift than an object B, using the
same baseline, then A is closer than B.
 Extension: A longer baseline will produce a
larger parallax shift for the same object.
How to measure d1 or d2?
If the scale were correct in the diagram, we would use
“triangulation”
Measure the baseline AB.
Measure the angles at A and B.
Create a simple scale drawing and measure d.
OR, use simple trigonometry to calculate d.
But…….in astronomy:



The distances to most objects of interest
are extremely large.
Earth-based baselines make angles at A
and B nearly 90 degrees!
These angles are nearly impossible to
measure accurately.
The picture becomes…….
….a very long, thin isosceles triangle.
The issue becomes one of measuring:
VERY TINY ANGLES!!
The Small Angle Formula
D
A
d
The Small Angle formula becomes:
Angle A = (D/d) x 206,265
(A in arcseconds)
Outdoor Parallax
C
“Pole”
p’
B
Distant
Background
p
Baseline (b)
A
d
D
p = parallax angle
Basic Geometry
A
p
B
d
The small angle formula gives us: p = (AB/d) x 57.3
OR (with a little algebra manipulation):
d = (AB/p) x 57.3 (p in degrees)
d = (AB/p) x 206,265 (p in arcsecs)
Outdoor approximation

It’s impossible to measure p directly. However,
if the distance to the background is >>> d, then
angle p’ is approximately equal to p.

We can measure p’ just as we have measured
angular size….by sighting on points C and D
with a ruler and a meter stick.
Measuring Angular Size
A
x
Eye
y
Picture on
wall
B
Hold ruler in front of your eye. Match up “x” with A
and B. Measure x. Partner measures y.
Angular size of picture = (X/Y) * 57.3 degrees
Example

Measure AB = 8.4 meters

Measure angle p’ (angular size of CD) using
ruler and meter stick. x = 5 cm, y = 60 cm.

Calculate p’ = x/y * 57.3 degrees = 4.8 deg.

Calculate d = 8.4/4.8 * 57.3 = 100 meters.
The “Parallax Formula”
The formula:
d = (AB/p”) x 206,265
works for ALL applications of astronomical
parallax, where p” is the parallax angle in
arcsecs. d will have the same units as AB.
Example
Suppose the Yerkes telescope in Wisconsin and the
Leuschner telescope near San Francisco take an
image of the same asteroid, at the same time.
They measure a parallax angle of 4 arcsecs (4”).
The baseline, AB, is the distance between the two
scopes = 3200 km.
d = (3200/4) x 206265 km = 165 million km.
(For reference: the Earth-Sun distance is about 150
million km.)
Making the baseline longer:
The “2 AU baseline”
Earth-Dec
1 AU
p”
p”
Sun
1 AU
1 AU (Astronomical Unit) =
Earth-Sun distance = 150 million km
Earth-June
Parallax with 2 AU baseline
Plug into the Parallax formula:
d = (AB/p”) x 206,265
d = (2 AU/2p”) x 206,265
d = 3.1 E16/p” meters
Very large numbers…..need a new unit!!
The Parsec
Definition: the distance that results when an
object has a parallax angle of 1 arcsec with
a baseline of 1 AU. The word, “parsec”,
comes from a combination of “parallax”
and “arcsecond”.
So, 1 parsec (pc) = 3.1 E16 meters = 31
trillion km.
Light Year vs. Parsec
One Light Year: the distance light travels in
one year = 9.5 E15 meters = 9.5 trillion km.
One parsec = 31 trillion km.
So, one parsec = 3.26 light years
A simple formula
When astronomers observe the parallax of
stars using the 2 AU baseline, then the
parallax formula becomes:
d = 1/p”
And d is always in units of parsecs.
Examples: p” = 4 arcsecs, d = 0.25 pc.
p” = 0.2 arcsecs, d = 5 pc.
Min parallax = Max distance

The Hipparcos satellite can measure
parallax angles to around 1 milliarcsec =
0.001”
 Maximum distance = 1/0.001” =
1000 pc
or about 3300 ly (light years)
How to measure p?
In astronomy we measure the parallax shift
directly from two images taken by two
different telescopes.
See example of asteroid “Austria” using
Hands-On Universe-Image Processing
Digital Images

Each image is made of pixels

Each pixel represents a certain angle in
space

Given by the Plate Scale in “/pixel.
Example
Plate Scale = 0.8”/px.
Measure the shift in pixels. Suppose its 20 pixels.
Then p” = 20 x 0.8 = 16”
Distance calculation
Use the parallax formula:
d = (AB/p”) x 206,265
For p” = 16” and AB = 6,000 km. (e.g.)
d = (6,000/16) x 206,265 = 7.7 E7 km = .77 AU